My question is obviously based on the title. I want to show that there is no retraction of a $3$-sphere (denoted $S^3$) onto the torus $T^2$ (doughnut surface). Any ideas on how one should do this? Input would be highly appreciated.

Jacob Schlather's answer is probably the best. But, if you wanted a purely topological reason, you could use the fact that a $T^2$ in $S^3$ must bound a solid torus on at least one side. Then a retract of $S^3$ onto $T^2$ would induce a retract of $D^2\times S^1$ onto $T^2$, which is impossible (this last statement may require some justification).